Rank four | History | Varuna | Octowerck

Rank four

These are rank three temperaments tempering out 441/440, but we can began by looking at the rank four temperament.

Minimax tuning
[|1 0 0 0 0>, |1/2 2/3 1/6 -1/3 1/6>, |0 0 1 0 0>,
|1 -2/3 1/3 1/3 1/3>, |0 0 0 0 1>]
Eigenmonzos: 2, 5, 11, 9/7

Map: [<1 0 0 0 -3|, <0 1 0 0 2|, <0 0 1 0 -1|, <0 0 0 1 2|]
EDOs: 72, 118, 130, 159, 171, 190, 248, 289, 320

History

Commas: 441/400, 4000/3993

Minimax tuning
[|1 0 0 0 0>, |4/17 12/17 0 -6/17 6/17>,
|-27/17 4/17 0 -2/17 19/17>,
|8/17 -10/17 0 5/17 12/17>, |0 0 0 0 1>]
Eigenmonzos: 2, 11, 9/7

Map: [<1 2 0 0 1|, <0 -6 0 7 2|, <0 0 1 1 1|]
Generators: 2, 21/20, 5
EDOs: 58, 72, 130, 202, 217, 289, 491
Badness: 0.000604

13-limit

Commas: 364/363, 441/440, 1001/1000

Minimax tuning
[|1 0 0 0 0 0>, |5/4 0 0 -3/8 0 3/8>, |0 0 1 0 0 0>,
|7/8 0 1 7/16 0 -7/16>, |5/4 0 1 1/8 0 -1/8>,
|7/8 0 1 -9/16 0 9/16>]
Eigenmonzos: 2, 5, 13/7

Map: [<1 2 0 0 1 2|, <0 -6 0 7 2 -9|, <0 0 1 1 1 1|]
Generators: 2, 21/20, 5
EDOs: 58, 72, 87, 130, 217, 289
Badness: 0.000540

Spectrum: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7

Varuna

Commas: 441/440, 8019/8000

Map: [<2 0 0 9 12|, <0 1 0 -4 -6|, <0 0 1 2 3|]
EDOs: 12, 26, 46, 58, 72, 118, 130, 248, 320
Generators: 99/70, 3, 5
Badness: 0.000419

Octowerck

Commas: 420175/419904, 321489/320000

POTE generator: ~350/243 = 633.738

Map: [<8 0 -11 14|, <0 3 7 2|]
EDOs: 72, 176, 248, 320
Badness: 0.1023

11-limit

Commas: 441/440, 8019/8000, 41503/41472

POTE generator: ~121/84 = 633.355

Map: [<8 0 -11 14 15|, <0 3 7 2 3|]
EDOs: 72, 176, 248, 320
Badness: 0.0302

13-limit

Commas: 441/440, 729/728, 1001/1000, 6656/6655

POTE generator: ~75/52 = 633.742

Map: [<8 0 -11 14 15 -38|, <0 3 7 2 3 16|]
EDOs: 72, 248, 320
Badness: 0.0276